Andrew Carnie. Syntax A Generative Introduction (3 ed, 2012). p 209.
Pls see red underline. I don't see what is problem?
The problem is that the determiner itself doesn't act like a phrase. For example, you can't conjoin two determiners (*the and my book), you can't attach complements to determiners, and so on. We could say that the determiner is a special type of phrase which acts differently from all the others, but that introduces ugly exceptions into the theory—it's much more elegant if all phrases act the same way. So we can't call "the" a phrase.
But on the flipside, we also want each phrase to contain only a single head, plus any number of other phrases. We know that the noun (N) acts as the head in a noun phrase (NP), so we'd really like everything else inside the NP to be a phrase of some sort.
The "DP hypothesis", as the book calls it, gets around this by making the determiner the head of a phrase which contains the entire NP as its complement. This means determiner phrases (DPs) act the same as other phrases in terms of conjunction, complements, and so on.